### 数学代写|黎曼几何代写Riemannian geometry代考|MAST90143

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼几何Riemannian geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼几何Riemannian geometry代写方面经验极为丰富，各种代写黎曼几何Riemannian geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼几何代写Riemannian geometry代考|Negative Curvature: The Hyperbolic Plane

The negative constant curvature model is the hyperbolic plane $H_{r}^{2}$ obtained as the surface of $\mathbb{R}^{3}$, endowed with the hyperbolic metric, defined as the zero level set of the function
$$a(x, y, z)=x^{2}+y^{2}-z^{2}+r^{2} .$$
Indeed, this surface is a two-fold hyperboloid, so we can restrict our attention to the set of points $H_{r}^{2}=a^{-1}(0) \cap{z>0}$.

In analogy with the positive constant curvature model (which is the set of points in $\mathbb{R}^{3}$ whose Euclidean norm is constant) the negative constant curvature model can be seen as the set of points whose hyperbolic norm is constant in $\mathbb{R}^{3}$. In other words,
$$H_{r}^{2}=\left{q=(x, y, z) \in \mathbb{R}^{3} \mid|q|_{h}^{2}=-r^{2}\right} \cap{z>0}$$
The hyperbolic Gauss map associated with this surface can be easily computed, since it is explicitly given by
$$\mathcal{N}: H_{r}^{2} \rightarrow H^{2}, \quad \mathcal{N}(q)=\frac{1}{r} \nabla_{q} a$$
Exercise 1.63 Prove that the Gaussian curvature of $H_{r}^{2}$ is $\kappa=-1 / r^{2}$ at every point $q \in H_{r}^{2}$.

We can now discuss the structure of geodesics and curves with constant geodesic curvature on the hyperbolic space. We start with a result that can be proved in an analogous way to Proposition $1.60$. The proof is left to the reader.
Proposition 1.64 Let $\gamma:[0, T] \rightarrow H_{r}^{2}$ be a curve with unit speed and constant geodesic curvature equal to $c \in \mathbb{R}$. For every vector $w \in \mathbb{R}^{3}$, the function $\alpha(t)=\langle\dot{\gamma}(t) \mid w\rangle_{h}$ is a solution of the differential equation
$$\ddot{\alpha}(t)+\left(c^{2}-\frac{1}{r^{2}}\right) \alpha(t)=0 .$$

## 数学代写|黎曼几何代写Riemannian geometry代考|Tangent Vectors and Vector Fields

Let $M$ be a smooth $n$-dimensional manifold and let $\gamma_{1}, \gamma_{2}: I \rightarrow M$ be two smooth curves based at $q=\gamma_{1}(0)=\gamma_{2}(0) \in M$. We say that $\gamma_{1}$ and $\gamma_{2}$ are equivalent if they have the same first-order Taylor polynomial in some (or, equivalently, in every) coordinate chart. This defines an equivalence relation on the space of smooth curves based at $q$.

Definition 2.1 Let $M$ be a smooth $n$-dimensional manifold and let $\gamma: I \rightarrow$ $M$ be a smooth curve such that $\gamma(0)=q \in M$. Its tangent vector at $q=\gamma(0)$, denoted by
$$\left.\frac{d}{d t}\right|_{t=0} \gamma(t) \quad \text { or } \quad \dot{\gamma}(0),$$
is the equivalence class in the space of all smooth curves in $M$ such that $\gamma(0)=$ $q$ (with respect to the equivalence relation defined above).

It is easy to check, using the chain rule, that this definition is well posed (i.e., it does not depend on the representative curve).

Definition $2.2$ Let $M$ be a smooth $n$-dimensional manifold. The tangent space to $M$ at a point $q \in M$ is the set
$$T_{q} M:=\left{\left.\frac{d}{d t}\right|{t=0} \gamma(t) \mid \gamma: I \rightarrow M \text { smooth, } \gamma(0)=q\right} .$$ It is a standard fact that $T{q} M$ has a natural structure of an $n$-dimensional vector space, where $n=\operatorname{dim} M$.

Definition 2.3 A smooth vector field on a smooth manifold $M$ is a smooth map
$$X: q \mapsto X(q) \in T_{q} M$$
that associates with every point $q$ in $M$ a tangent vector at $q$. We denote by $\operatorname{Vec}(M)$ the set of smooth vector fields on $M$.

In coordinates we can write $X=\sum_{i=1}^{n} X^{i}(x) \partial / \partial x_{i}$, and the vector field is smooth if its components $X^{i}(x)$ are smooth functions. The value of a vector field $X$ at a point $q$ is denoted, in what follows, by both $X(q)$ and $\left.X\right|_{q}$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Flow of a Vector Field

Given a complete vector field $X \in \operatorname{Vec}(M)$ we can consider the family of maps
$$\phi_{t}: M \rightarrow M, \quad \phi_{t}(q)=\gamma(t ; q), \quad t \in \mathbb{R}{2}$$ where $\gamma(t ; q)$ is the integral curve of $X$ starting at $q$ when $t=0$. By Theorem $2.5$ it follows that the map $$\phi: \mathbb{R} \times M \rightarrow M{,} \quad \phi(t, q)=\phi_{t}(q)$$
is smooth in both variables and the family $\left{\phi_{t}, t \in \mathbb{R}\right}$ is a one-parametric subgroup of Diff $(M)$; namely, it satisfies the following identities:
\begin{aligned} \phi_{0} &=\mathrm{Id}{+} \ \phi{t} \circ \phi_{s} &=\phi_{s} \circ \phi_{t}=\phi_{t+s}, \quad \forall t, s \subset \mathbb{R}, \ \left(\phi_{t}\right)^{-1} &=\phi_{-t}, \quad \forall t \in \mathbb{R} . \end{aligned}

Moreover, by construction, we have
$$\frac{\partial \phi_{t}(q)}{\partial t}=X\left(\phi_{t}(q)\right), \quad \phi_{0}(q)=q, \quad \forall q \in M$$
The family of maps $\phi_{t}$ defined by $(2.5)$ is called the flow generated by $X$. For the flow $\phi_{t}$ of a vector field $X$ it is convenient to use the exponential notation $\phi_{t}:=e^{t X}$, for every $t \in \mathbb{R}$. Using this notation, the group properties (2.6) take the form
$$\begin{gathered} e^{0 X}=\mathrm{Id}, \quad e^{t X} \circ e^{s X}=e^{s X} \circ e^{t X}=e^{(t+s) X}, \quad\left(e^{t X}\right)^{-1}=e^{-t X} \ \frac{d}{d t} e^{t X}(q)=X\left(e^{t X}(q)\right), \quad \forall q \in M \end{gathered}$$
Remark $2.8$ When $X(x)=A x$ is a linear vector field on $\mathbb{R}^{n}$, where $A$ is an $n \times n$ matrix, the corresponding flow $\phi_{t}$ is the matrix exponential $\phi_{t}(x)=e^{t A} x$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Negative Curvature: The Hyperbolic Plane

H_{r}^{2}=\left{q=(x, y, z) \in \mathbb{R}^{3} \mid|q|_{h}^{2}=-r^{ 2}\right} \cap{z>0}H_{r}^{2}=\left{q=(x, y, z) \in \mathbb{R}^{3} \mid|q|_{h}^{2}=-r^{ 2}\right} \cap{z>0}

ñ:Hr2→H2,ñ(q)=1r∇q一个

## 数学代写|黎曼几何代写Riemannian geometry代考|Tangent Vectors and Vector Fields

dd吨|吨=0C(吨) 或者 C˙(0),

T_{q} M:=\left{\left.\frac{d}{d t}\right|{t=0} \gamma(t) \mid \gamma: I \rightarrow M \text { smooth, } \伽马(0)=q\right} 。T_{q} M:=\left{\left.\frac{d}{d t}\right|{t=0} \gamma(t) \mid \gamma: I \rightarrow M \text { smooth, } \伽马(0)=q\right} 。一个标准的事实是吨q米有一个自然的结构n维向量空间，其中n=暗淡⁡米.

X:q↦X(q)∈吨q米

## 数学代写|黎曼几何代写Riemannian geometry代考|Flow of a Vector Field

φ吨:米→米,φ吨(q)=C(吨;q),吨∈R2在哪里C(吨;q)是积分曲线X开始于q什么时候吨=0. 按定理2.5随之而来的是地图

φ:R×米→米,φ(吨,q)=φ吨(q)

φ0=我d+ φ吨∘φs=φs∘φ吨=φ吨+s,∀吨,s⊂R, (φ吨)−1=φ−吨,∀吨∈R.

∂φ吨(q)∂吨=X(φ吨(q)),φ0(q)=q,∀q∈米

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。